Effect of the Domain Geometry on the Solutions to Fractional Brezis-Nirenberg Problem

Abstract

In this paper, we consider the Brezis-Nirenberg problem for the nonlocal fractional elliptic equation , , , , , , where is fixed, , is a small parameter, and is a bounded smooth domain of . denotes the fractional Laplace operator defined through the spectral decomposition. Under some geometry hypothesis on the domain , we show that all solutions to this problem are least energy solutions.

1. Introduction and Main Results

In the famous paper of Brezis and Nirenberg [1], they studied the following nonlinear critical elliptic partial differential equation:where is a bounded domain in . They proved that problem (1) has a positive nontrivial solution provided and , where is the first eigenvalue of in . This result was extended by Capozzi et al. [2] for every parameter . Rey [3] and Han [4] established the asymptotic behavior of positive solutions to problem (1) by different methods independently.

In this paper, we study the following nonlocal Brezis-Nirenberg problem:where and , , is a bounded smooth domain of , and is the spectral fractional Laplacian defined in terms of the spectra of the in ; for more details see Section 2.

The qualitative properties of solutions to problem (2), such as existence, nonexistence, and multiplicity results, were widely studied; see [5–11] and references therein. It is well-known that [5] problem (2) has at least one positive solution for each small . On the other hand, when is a domain, by the Pohozaev identity [12, 13], we know that problem (2) does not have any solutions in a star-shaped domain when . Consequently, the solutions to problem (2) blow up at some points as . Therefore, as , there exist subsequences , , and a point , such that and as . In the following, we mainly consider the solution .

Based on fractional harmonic extension formula of Caffarelli and Silvestre [14] (see Cabré and Tan [15] also), Choi et al. [9] studied the asymptotic behavior of least energy solutions to problem (2); that is, satisfieswhere is the best constant in fractional Sobolev inequality. For some other related results, see [6, 8, 16] and references therein.

The main goal of this paper is to show that, under some hypothesis on the domain , all solutions to problem (2) automatically satisfy (3); that is, all solutions are least energy solutions.

Our main result is the following.

Theorem 1. Let be a smooth bounded domain of , which is symmetric with respect to the coordinate hyperplanes and convex in the directions for . Then all solutions to problem (2) are least energy solutions.

Remark 2. According to Theorem 1.3 in [9], we know that there exist a point and a family of solutions to (2), which blow up and concentrate at the point as . Without loss of generality we assume that in this paper.

Remark 3. This results are motivated by the work of Cerqueti and Grossi [17] about the classical Brezis-Nirenberg problem In [17], they obtained the asymptotical behavior of any solution to the above equation in a neighborhood of the origin. Furthermore, uniqueness and nondegeneracy result for the solutions also obtained.

The paper is organized as follows. Section 2 contains some notations and definitions. Section 3 is concerned with the proof of Theorem 1.

2. Useful Definitions

First of all, in this section we recall some basic properties of the spectral fractional Laplacian.

In this paper, the letter will denote a positive constant, not necessarily the same everywhere. Let be a smooth bounded domain of . , are the eigenvalues of the Dirichlet Laplacian on , and are the corresponding normalized eigenfunctions; namely, Define the fractional Laplacian aswhere fractional Sobolev space is defined as

It is interesting to note that another very popular “integral” fractional Laplacian is defined asup to a normalization constant which will be omitted for brevity. For differences between the spectral fractional Laplacian (6) and the fractional Laplacian (8), see [18–20].

In this paper, we mainly consider some properties of isolated blow-up point.

Definition 4. Suppose that is a solution to problem (2). The point is called a blow-up point of , if there exists a sequence of point , such that and .

The concept of an isolated blow-up point was first introduced by Schoen; for more details of the definition of isolated blow-up point, see [17, 21].

Definition 5. Let be a solution to problem (2). A point is an isolated blow-up point of if there exist , and a sequence , such that is a local maximum point of , , and for any

3. Proof of Theorem 1

In this section, we give the proof of Theorem 1, which will be divided into three lemmas.

Firstly, according to Remark 2, we know that is the blow-up point; in view of Definition 4, there exists a sequence of point and , such that and . In the following, the index is omitted for the sake of simplicity. The following lemma shows that is an isolated blow-up point.

Lemma 6. Let be a solution to problem (2). Then there exists a constant such that

Proof. Suppose, on the contrary, there exists a , such thatAccording to Lemma 3.1 in [9], there is a constant such that , where This fact implies that . Define By Lemma 3.3 in [9], we know that, up to a subsequence, converges to the function uniformly on any compact set, where Obviously, ; then for fixed , , which implies that . Note that in for and is the extreme point of . Then there exists a sequence of points , such that . Taking into account Remark 2, we find This fact, together with (11), implies that which contradicts the fact that . This proves the validity of this lemma.

Lemma 7. Let be a solution of problem (2). Then there exist two positive constants and , such that

Remark 8. It can be easily seen that uniformly for .

Proof. Analysis similar to that in the proof of Proposition 4.9 in [21] shows that there exists a positive constant such that, for any , This fact implies that there exist and such that for any Particularly, Now we argue by contradiction to show (17) holds. Suppose that there exists a point , say the maximum point of in , such that It can be easily seen that . Thus . Consequently, ; this contradicts the fact that .

Remark 9. It is worth pointing out that, in [21], Jin et al. consider the integral fractional Laplacian, but the arguments of Propositions 4.4 and 4.9 in [21] still hold for spectral fractional Laplacian.

Lemma 10. Let be a solution to problem (2). Then

Proof. Decompose as , where and ; is the constant which appears in Lemma 7; we get By (17), it is obvious that . On the other hand, for any , define By a similar argument as Proposition 4.4 in [21], we derive that in . Therefore, by the dominated convergence theorem, we find This finishes the proof of (22).

Proof of Theorem 1. Since is the solution to problem (2), thereforeAccording to the Hölder inequality, we findThus, taking into account (26) and (27), we obtainThis fact, combined with (22), shows that (3) holds. This completes the proof of Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors equally contributed in this work.

Acknowledgments

This work is partially supported by the National Science Foundation of China (No. 11761059), General Topics of National Education and Scientific Research (No. ZXYB18019), Fundamental Research Funds for the Central Universities (No. 31920170001), and Key Subject of Gansu Province.